Math Word Problem- Multiple Choice

Question

There are seven clarinet players in the concert band. In how many ways can they be seated and seven chairs at a concert? Remember you can’t have the same person sitting in more than 1 chair.

A- 5,040

B- 2,520

C - 840

D - 210

Answer 1

Hi Kalise N.,

This is a fairly standard problem -- except for what it doesn't say!

So, this is a "permutations" type of problem, because, you want to know exactly which person sits in which seat. To think of how to do this, imagine that all the clarinet players are standing off on the side, and YOU are responsible for seating them! So, into the first chair (first, in the sense that you are keeping track of the chairs), you may put any of the 7. That's 7 choices. Then, into the next chair, you may seat any of the remaining 6 players; that's 6 ways of filling the second chair AFTER you've filled the first chair. Similarly, the third chair can be filled by any of the remaining 5 players, and so on down to the last chair which can ONLY be filled by the last remaining player, whoever that is. So the total number of ways in which those 7 ordered seats may be filled is, 7 x 6 x 5 x 4 x 3 x 2 x 1 = 7! or 5040.

Now for some fun! Suppose there were 7 places for clarinet chairs on the stage, and 7 distinctive chairs, and 7 (obviously distinct) clarinet players -- how many ways are now possible? To track that, once you have your 5040 possibilities for matching players to chairs, the process repeats for putting the chairs onto the stage -- so that's 5040 x 5040 = 25,401,600 ways.

Now suppose that you could mix the clarinet chairs with every other instrument's, on the stage -- well, you can imagine that soon you would run out of ink calculating the possibilities! Perhaps you can see why everyone sits in their own fixed place in the orchestra -- (let alone the problems of the conductor in cueing particular instrument sections during the performance, otherwise)!

Hope this helps you,

-- Cheers, -- Mr. d.